Extremal behavior of diffusion models in finance
type: Item
instance of: scholarly article
title: Extremal behavior of diffusion models in finance
zbMATH Open document ID: 0931.60036
DOI: 10.1023/A:1009961817149
author: Milan Borkovec
author: Claudia Klüppelberg
published in: Extremes
publication date: 1999-08-10T00:00:00Z
review text: The authors study the extremal behavior of diffusion processes \(X_{t}\) defined by SDE \(dX_{t}=\mu(X_{t})dt+\sigma(X_{t})dw(t),\) with \(t>0\) and \(X_{0}=x,\) where \(w\) is a standard Brownian motion, \(\mu\) is the drift term and \(\sigma\) is the diffusion coefficient or volatility. Extremal behavior of a stochastic process \(X_{t}\) is for instance manifested in the asymptotic behavior of the running maxima \(M^{X}_{t}=\max_{0\leq s\leq t}X_{s},\;t>0,\) or running minima. The authors restrict themselves to the investigation of maxima, the mathematical treatment for minima being similar. The authors furthermore investigate the point process of \(\varepsilon\)-upcrossing of a high threshold \(u\) by \(X_{t}.\) For fixed \(\varepsilon>0\) the process has an \(\varepsilon\)-upcrossing at \(t\) if it has remained below \(u\) on the interval \((t-\varepsilon,t)\) and is equal to \(u\) at \(t.\) Under the weak conditions the point process of \(\varepsilon\)-upcrossing, properly scaled in time and space, converges in distribution to a homogeneous Poisson process. Two standard models like above-mentioned SDE in finance are the Black-Scholes model and the Vasicek model. The paper also aims at applications in finance: for the various models, namely Vasicek model, Cox-Ingersoll-Ross model and its generalized version, the generalized hyperbolic diffusion, the authors derive the distributional behavior of \(M^{X}_{t}\) as \(t\to+\infty\) which describes the extremal behavior of the whole process \(X_{t}.\) A new model, the generalized inverse Gaussian diffusion, is introduced. The results of this paper can be applied to study the risk measures of financial products as for instance the value of risk or related quantile risk measures.
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J70
Mathematics Subject Classification ID: 60G70
zbMATH DE Number: 1323309
zbMATH Keywords: extreme value theory
zbMATH Keywords: Cox-Ingersoll-Ross model
zbMATH Keywords: generalized inverse Gaussian diffusion
zbMATH Keywords: generalized hyperbolic diffusion
zbMATH Keywords: \(\varepsilon\)-upcrossings
zbMATH Keywords: Poisson approximation
zbMATH Keywords: price process
zbMATH Keywords: recurrent diffusion
zbMATH Keywords: term structure model
zbMATH Keywords: Vasicek model
zbMATH Keywords: volatility
MaRDI profile type: Publication
full work available at URL: a:1009961817149
OpenAlex ID: W2293539173
description: scientific article; zbMATH DE number 1323309